You have fine-grained control over who can access each Live Chat channel. Navigate to Settings → Channels → Chat & Social → [your chat] → Teammates Management to change your settings. The following options are available:

• Only me Only you yourself can view the Live Chat channel.
• Everyone All agents can view the Live Chat channel.
• Selected agents Only the specified agents can view the Live Chat channel. After selecting this option a list of all agents in your organization will be displayed, and you can select one or more agents. When no agents are selected the setting will revert back to "Everyone".
• Selected teams Only the specified teams can view the Live Chat channel. After selecting this option a list of all teams in your organization will be displayed, and you can select one or more teams. When no teams are selected the setting will revert back to "Everyone".